Word for "number of times a value is copied"

Maybe propagation depth, in reference to how many generations a trait gets propagated.

Other terms that might sound less biology-related could be copy count, value copy count, iterations.

You might consider using multiplicity. In mathematics this refers to the number of times the same root of an equation or zero of a function occurs.

Value repetition could convey the meaning you want.

I'd use a title like data age or data vintage. The latter usage would be analogous to vintage denoting the "year and vineyard or district of origin" of a wine.

Incidentally, it probably is inappropriate to say that "the values become less reliable because of genetic changes". The data does not become less reliable, it becomes less applicable to current generations.

Update: Phrase copy count is short and to the point. Also replications.

I just came up with preservation times, maintenance times, and persistence times for discussion.